Theorem submgmmgm

Description: Submagmas are themselves magmas under the given operation. (Contributed by AV, 26-Feb-2020)

	Ref	Expression
Hypothesis	submgmmgm.h	⊢ 𝐻 = ( 𝑀 ↾s 𝑆 )
Assertion	submgmmgm	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝐻 ∈ Mgm )

Proof

Step	Hyp	Ref	Expression
1		submgmmgm.h	⊢ 𝐻 = ( 𝑀 ↾s 𝑆 )
2		submgmrcl	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝑀 ∈ Mgm )
3		eqid	⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 )
4	3 1	issubmgm2	⊢ ( 𝑀 ∈ Mgm → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ 𝐻 ∈ Mgm ) ) )
5	2 4	syl	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ 𝐻 ∈ Mgm ) ) )
6	5	ibi	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ 𝐻 ∈ Mgm ) )
7	6	simprd	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝐻 ∈ Mgm )